Trigonometry can be somewhat complicated when you start studying. However, as we advance in their knowledge, everything seems to make sense. In this article we want to help you to get to know trigonometry and the trigonometric ratio or relationship a little better. In this way we want to contribute our own grain of sand to your training. You will see how, once you finish reading the article, it will be much easier for you to understand these concepts.
In the next few lines you will learn what trigonometry is. Also what is the trigonometric ratio and what is its meaning. Soon you will be able to explain this concept to your friends and brothers and teach it yourself. Are you ready to learn new things about this interesting field? In that case, keep reading!
What is trigonometry? Its meaning
Did you know that the term trigonometry comes from the Greek? Well, that’s how it is and there are several words that give life to this term. Let’s see what they are:
- Trigonon meaning triangle.
- Metron, which means to measure.
- Tria, which means three.
Taking into account that the term trigonometry is made up of these three words, it is not difficult to understand what this specialty of mathematics is dedicated to. Trigonometry is that specialty that calculates the elements of triangles. It also carries out all the calculations associated with triangles. Both between its sides and between its angles.
To make these complex calculations, trigonometry uses three basic units of measurement. These units of measurement are the following:
- The radian. The radian is the natural unit of angles. According to this unit any circumference can be divided into 2 π radians. Specifically, it is the measure of a central angle whose arc has a length identical to the radius of the circumference. The radian is expressed under the symbol rad.
- The sexagesimal degree. It is used when a circumference is divided into a total of 360º sexagesimals.
- The centesimal degree or gradian. The centesimal degree or gradian also divides a circle. However, unlike the sexagesimal degree, which divides it into 360º, the gradian, or centesimal degree, does so into 400º centesimals.
What are the ratios or trigonometric relationship. Its meaning?
Now that we know what trigonometry studies, it is time to understand what trigonometric ratios or relationships are. Etymologically speaking this term is composed of two words.
- Reasons. The word reasons derives from the Latin term ratio whose meaning is precisely reason.
- Trigonometric. The word trigonometric means relating to trigonometry. Unlike reasons, as we have seen a few lines above, the term trigonometry comes from the fusion of several Greek terms. Specifically 2: trigonon and In addition it is accompanied by the suffix -ico. This suffix means relative to.
The trigonometric ratios are the relationships that are created between the different sides that make up a 90º triangle. These reasons can be three.
- The sine is the ratio or relationship that exists between the hypotenuse and the opposite leg.
- We speak of cosine to refer to the ratio or relationship that exists between the hypotenuse and the leg adjacent to it.
- Finally, when talking about tangent, we refer to the ratio or relationship that exists between the opposite leg and its adjacent leg.
It is possible that these explanations do not end up making sense to you because you do not know which are the legs, opposite and adjacent, or the hypotenuse. Let’s take a look at these terms below.
- Leg. When talking about leg we refer to the smaller sides that make up a right triangle. These sides are what create the 90º angle or right angle. These legs are also known as sine and cosine.
- Hypotenuse. The longest side of the triangle, which in turn is opposite the 90º angle or right angle, is what we know as the hypotenuse.
Although the sine, cosine and tangent are the most used trigonometric ratios or relationships, there are others than this branch of mathematics as well. And, although less known, it is important to talk about them since they are also trigonometric ratios. These other trigonometric ratios or relationships are:
- The secant. When we talk about the secant we do it to refer to the ratio or relationship that exists between the adjacent leg and the hypotenuse. You can also talk about the ratio or reciprocal relationship of the cosine.
- The cosecant. When speaking of the cosecant, reference is made to the relationship or ratio that exists between the opposite leg and the hypotenuse. You can also talk about the ratio or reciprocal relationship of the sine.
- The cotangent. If you hear about the cotangent, it is because reference is being made to the ratio or relationship between both legs. That is, the opposite leg and the adjacent leg. You can also talk about the ratio or reciprocal relationship of the tangent.
Some exercises of ratios or trigonometric relationships
Now that we know what trigonometric relationships are, we are going to propose some exercises for you to apply this knowledge that you have learned. Let’s review them before we start with them:
Since this is the reason we have to:
The secant of α = Secα = hypotenuse / adjacent leg. You can see it in the following illustration.
The cosecant of α = Csecα = hypotenuse / opposite leg. You can more easily study the formula in the following image.
Now that we have laid the foundations, it is time to start the exercises.
Trigonometric relations exercises
- Exercise 1. We are going to offer you the data related to the hypotenuse, the opposite leg and the adjacent leg. The exercise will consist of calculating what the secant is. You dare? In that case take a look at the illustration. If you have doubts about how to solve it, you can see the solution a few lines below. Good luck with the exercise!
- Exercise 2.In this case we want you to find out the cosecant. For this we offer you the data related to the legs (the opposite leg and the adjacent leg) and the hypotenuse. You can check them in the following illustration. If you are not sure how to solve this exercise, you can take a look at the solution a few lines below. Try it by applying the formulas we just reviewed. It sure doesn’t cost you anything!
Solutions to the exercises of trigonometric relations
- Solution to exercise 1. In this exercise we asked you to find out what the secant was. Have you found out what the solution was? Are you not clear? In that case take a look at the answer. You can find out in the illustration below these lines.
- Solution to exercise 2. In this exercise the idea was to find out what the cosecant of this triangle was. For this we provide you with the data of the legs and the hypotenuse. If you want to check if you found the correct solution, be sure to take a look at the answer. You have it in the following illustration.
How have you been given these two exercises? Was it difficult or easy for you? You can do many more to control these two formulas as much as possible. Tell your parents about it so they can do special exercises for you. Once you’ve gotten the hang of it, you can create them yourself and have competitions with your friends to see who can solve them faster. You will have a lot of fun and you will also learn.